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Math Help - help me in solving this

  1. #1
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    help me in solving this

    Find a closed form expression (i.e., a rational function) for the generating function for the number of ways to make n cents out of nickels, dimes and quarters if you must use an even number of nickels, an odd number of dimes, at least 2 quarters, at most 20 nickels, and at most 12 dimes? Expand the expression showing all nonzero terms through x100. Note: The number of such ways for 80 cents and 85 cents is 2 and 1, respectively
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    Quote Originally Posted by kkkkkk View Post
    Find a closed form expression (i.e., a rational function) for the generating function for the number of ways to make n cents out of nickels, dimes and quarters if you must use an even number of nickels, an odd number of dimes, at least 2 quarters, at most 20 nickels, and at most 12 dimes? Expand the expression showing all nonzero terms through x100. Note: The number of such ways for 80 cents and 85 cents is 2 and 1, respectively
    I'll try to get you started.

    The ordinary power series generating function for the value of an even number of nickels,at most 20, is
    \sum_{i=0}^{10} x^{5 \cdot 2i}

    for an odd number of dimes, at most 12 (i.e., at most 11)
    \sum_{j=0}^{5} x^{10 (2j+1)}

    for at least 2 quarters,
    \sum_{k=2}^\infty x^{25k}

    Multiply those three series together and you have the generating function for the number of ways to make change using the specified coins. You will have to sum the series involved in order to get your answer in the form of a rational function.
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